tools-only Abstract State Analyzer
Overview
This skill performs abstract interpretation to statically analyze source code and infer possible program states, variable ranges, and data properties. It identifies potential runtime errors without executing the program.
Analysis Workflow
Step 1: Parse and Understand Code Structure
Analyze the code to identify:
- Functions and their control flow
- Variable declarations and types
- Loops and conditionals
- Array/buffer operations
- Pointer/reference operations
- Function calls and parameter passing
Step 2: Select Abstract Domains
Choose appropriate abstract domains based on the analysis goals:
Interval Domain: Track numeric variable ranges
- Example:
x ∈ [0, 100]means x is between 0 and 100 - Good for: Array bounds checking, overflow detection
Sign Domain: Track whether values are positive, negative, or zero
- Values: {+, -, 0, ⊤}
- Good for: Division by zero, sign-dependent operations
Null Domain: Track whether pointers/references can be null
- Values: {null, not-null, maybe-null, ⊤}
- Good for: Null dereference detection
Type Domain: Track possible types of variables
- Good for: Type consistency checking, dynamic language analysis
Combination: Use multiple domains together for more precise analysis
Step 3: Initialize Abstract States
Set initial abstract values for:
- Function parameters (based on preconditions or ⊤ for unknown)
- Global variables
- Constants and literals
Example:
def process(arr, index):
# Initial state:
# arr: not-null (assumed)
# index: ⊤ (unknown integer)
Step 4: Perform Forward Analysis
Propagate abstract states through the program:
Assignment: Update abstract value
x = 5
# x: [5, 5]
y = x + 3
# y: [8, 8]
Conditionals: Split into branches
if x > 10:
# Branch 1: x ∈ [11, ∞]
else:
# Branch 2: x ∈ [-∞, 10]
Loops: Iterate until fixpoint
i = 0
while i < n:
# Iteration 1: i ∈ [0, 0]
# Iteration 2: i ∈ [0, 1]
# ...
# Fixpoint: i ∈ [0, n-1]
i += 1
Join Points: Merge states from multiple paths
if condition:
x = 5 # x: [5, 5]
else:
x = 10 # x: [10, 10]
# After join: x ∈ [5, 10]
Step 5: Detect Potential Errors
Check for violations at each operation:
Array Bounds:
arr[index]
# Check: index ∈ [0, len(arr)-1]?
# If index: ⊤ → Potential out-of-bounds
# If index: [0, 5] and len(arr) = 10 → Safe
Null Dereference:
ptr.field
# Check: ptr is not-null?
# If ptr: maybe-null → Potential null dereference
Division by Zero:
x / y
# Check: 0 ∉ y?
# If y: [1, 10] → Safe
# If y: [-5, 5] → Potential division by zero
Integer Overflow:
x = a * b
# Check: a * b within type bounds?
# If a: [1000, 2000], b: [1000, 2000] → Potential overflow for int32
Type Inconsistency:
result = func(arg)
# Check: arg type matches parameter type?
Step 6: Report Findings
For each potential error, report:
- Location: File, line number, function
- Error Type: Out-of-bounds, null dereference, etc.
- Abstract State: Variable values at the error point
- Severity: Definite error vs. potential error
- Explanation: Why the error might occur
- Suggestion: How to fix (add check, change bounds, etc.)
Complete Example
def find_max(arr, n):
if n <= 0:
return None
max_val = arr[0]
i = 1
while i < n:
if arr[i] > max_val:
max_val = arr[i]
i += 1
return max_val
Analysis:
Initial State:
- arr: not-null (assumed)
- n: ⊤ (unknown integer)
Line 2: if n <= 0
- Branch 1 (n ≤ 0): n ∈ [-∞, 0]
- Branch 2 (n > 0): n ∈ [1, ∞]
Line 3: return None (Branch 1)
- Safe return
Line 5: max_val = arr[0] (Branch 2)
- Access: arr[0]
- Check: 0 < len(arr)?
- POTENTIAL ERROR: arr length unknown, might be empty
- State: max_val = arr[0], n ∈ [1, ∞]
Line 6: i = 1
- State: i = [1, 1]
Line 7: while i < n
- Loop invariant: i ∈ [1, n]
- Fixpoint: i ∈ [1, n-1] inside loop
Line 8: if arr[i] > max_val
- Access: arr[i] where i ∈ [1, n-1]
- Check: i < len(arr)?
- POTENTIAL ERROR: If n > len(arr), out-of-bounds access
- State: max_val updated if arr[i] > max_val
Line 10: i += 1
- State: i ∈ [2, n]
Report:
POTENTIAL ERRORS FOUND:
1. Out-of-Bounds Access
Location: line 5, arr[0]
State: n ∈ [1, ∞], arr length unknown
Severity: Potential
Explanation: Array 'arr' might be empty when n > 0
Suggestion: Add check: if len(arr) == 0 or add precondition
2. Out-of-Bounds Access
Location: line 8, arr[i]
State: i ∈ [1, n-1], arr length unknown
Severity: Potential
Explanation: If n > len(arr), accessing beyond array bounds
Suggestion: Add precondition: n <= len(arr) or check i < len(arr)
Language-Specific Considerations
C/C++
- Track pointer arithmetic carefully
- Consider undefined behavior (signed overflow, null dereference)
- Analyze memory allocation/deallocation
- Check buffer sizes for string operations
Python
- Dynamic typing requires type domain
- List/dict operations need bounds checking
- None values require null domain
- Consider duck typing and attribute access
Java
- Null pointer exceptions
- Array bounds (ArrayIndexOutOfBoundsException)
- Integer overflow (silent wraparound)
- Type casting (ClassCastException)
JavaScript
- Undefined and null values
- Type coercion issues
- Array bounds (returns undefined, not error)
- Property access on null/undefined
Handling Complexity
Widening for Loops
When loops don't converge quickly, apply widening:
# Instead of: [0, 0] → [0, 1] → [0, 2] → ...
# Widen to: [0, ∞]
Function Summaries
For called functions, use summaries instead of full analysis:
def helper(x):
# Summary: returns x + 1, no errors
return x + 1
# Use summary instead of analyzing helper body
Path Sensitivity
For complex conditionals, track path conditions:
if x > 0 and x < 10:
# x ∈ [1, 9] (path-sensitive)
# vs x ∈ [-∞, ∞] (path-insensitive)
References
For detailed information on abstract interpretation techniques and domains:
- references/abstract_domains.md: Detailed abstract domain definitions and operations
- references/analysis_patterns.md: Common analysis patterns for different error types
- references/language_specifics.md: Language-specific analysis considerations